The Investigation on L 1 Adaptive Control of the TiltRotor Aircraft

College of Aerospace Engineering, Chongqing University, Chongqing 400044, China Science and Technology on Rotorcraft Aeromechanics Laboratory, China Helicopter Research and Development Institute, Jingdezhen, Jiangxi 333001, China National Key Laboratory of Science and Technology on Rotorcraft Aeromechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China


Introduction
A tilt-rotor aircraft has the advantages of both fixed-wing and rotorcraft such as high speed and long-range flight, vertical take-off, and landing ability. It can cover the flight envelope of conventional helicopters and propeller planes and has broad applications in both civil and military fields. However, due to the integration of flight modes and control modes of fixed-wing aircraft and helicopters, tilt-rotor aircraft has helicopter mode, conversion mode, and airplane mode, as shown in Figure 1. It is a typical morphing aircraft. Therefore, the mathematical model of flight dynamics is highly nonlinear. It is much more complicated in flight control system design compared with other aircraft.
Since the promulgation of ADS-33E-PRF, preliminary studies on flight control of tilt-rotor aircraft mainly focus on stability enhancement system design and quality improvement [1][2][3][4]. Juhasz developed a high order rotorcraft mathematical mode against a large civil tilt-rotor concept. At the same time, the corresponding control system is designed to meet the quality requirements [5]. In Ref. [6], neural network controller was used to accomplish the neural network path-tracking and real-time multitask flight simulation for the automatic transition of tilt-rotor aircraft. The flight performance of conversion mode is evaluated based on the linearized model. Rysdyk and Calise [7] propose an inverse control method based on neural network model to control civil tilt-rotor aircraft, which can ensure satisfied dynamic performance in the whole flight envelope. The stability of the system and the boundary of tracking error are strictly proved based on Lyapunov theory. Choi et al. [8] divide the nacelle angle into 90-60 degrees for helicopter mode, 60-30 degrees for conversion mode, and 30-0 degrees for airplane mode in the flight control system of unmanned tilt-rotor aircraft. The way of control scheduling is used, which is easy to be implemented in engineering. Lee et al. [9] divide the tiltrotor aircraft into different operating points based on discretization and optimize the control law parameters at different operating points using particle swarm optimization algorithm. In Ref. [10], based on an optimal control concept, an online optimization control method was developed for the tilt-rotor UAV. In Ref. [11], a smooth switching control scheme is provided for the tilt-rotor aircraft. However, the flight mechanic model in the above literature has been simplified to a large extent. As a result, the simulation results cannot reflect the real performance of the controllers. For a class of singularly perturbed Markov jump descriptor systems with nonlinear perturbation, a stochastic integral sliding mode control strategy is developed in [12,13]. In aeronautical engineering application field, the L 1 adaptive control has attracted wide attention in recent years. It is modified on the basis of the classical model reference adaptive control [14][15][16][17]. A low pass filter is introduced to decouple the adaptive and robustness of the control system, a larger adaptive gain can be adopted to improve the parameter estimation speed and obtain better control performance. The corresponding test work has been carried out, especially in the field of aerospace [18][19][20][21][22][23].
Motivated by the discussions above, we introduce the L 1 adaptive control method to the tilt-rotor aircraft in the flight control system. The major limitations for its applications can be summarized as follows: (1) The dynamics of tilt-rotor aircraft are usually highly nonlinear and coupled, and it has multiple flight modes. It is still a challenging problem to get desired dynamic performance for the controlled system in the full envelope (2) The performance of tilt-rotor aircraft control system should meet the related requirements in flight quality specification (3) The design of controller parameters should fully consider the physical realizability of the aircraft The paper is organized as follows. Section 2 introduces the mathematical model. Section 3 gives the L 1 adaptive controller design. Section 4 presents simulation studies. Finally, Section 5 draws the conclusion.   The XV-15 tilt-rotor aircraft uses both helicopter and airplane control strategies to control the aircraft. In helicopter flight mode, longitudinal cyclic, differential collective, and differential longitudinal cyclic are used to pitch, roll, yaw, and heave control, respectively. As the tilt-rotor aircraft converts from helicopter flight mode to airplane flight mode, the helicopter rotor control surfaces are washed out as a function of nacelle angle and flight speed.

Mathematical Model
The tilt-rotor aircraft flight dynamics model could be given by The state variables are given in the form of a vector X ∈ R 47×1 , which includes 26 rotor inflow states, 12 flapping motion states, and 9 aircraft states. u ∈ R 4×1 denotes a vector constituting the control inputs. The model detail can refer to Ref [24] (Ke Lu and C. S. Liu).

Trim Result Validation.
In the trim validation, the control inputs at different nacelle incidence angles are compared with simulation results from GTRS reports [25]. Figures 2  and 3, respectively, show the comparison of collective pitch and longitudinal control.
According to Figures 2 and 3, in helicopter mode, the variation trend of collective pitch is similar to the conventional helicopter. The collective pitch has the characteristic bucket profile as a function of flight speed. In airplane mode, the function of the rotor is providing forward pull force to overcome fuselage drag. That is to say, the wing is able to generate enough lift to overcome gravity. The collective pitch is much larger than that of the helicopter mode. In addition, longitudinal control input increases with speed in all flight modes.
Overall, the calculation results are quite consistent with GTRS results in all flight modes under the trim condition.

Linearized Result Validation.
With the application of a linearization algorithm, the rotor and inflow modes residualized out via quasistatic reduction, and then the nonlinear equation can be reduced to the form of To linearization validation, a comparison of the eigenvalues for matrix A in equation (2) with results from the flight test and GTRS is shown in Tables 2 and 3. As shown in Table 2, the characteristic in helicopter mode of the eigenvalue distribution between calculation results in this paper and flight test are very similar. In particular, for phugoid and dutch roll modes, calculation results in this paper and flight test have given the conclusion of instability. The airplane mode eigenvalues are shown in Table 3. In general, the calculation results in this paper are closer to GTRS, especially the prediction of short period modes.

Control Response Validation.
The control response verification is mainly to verify whether the response of tilt-rotor aircraft conforms to the laws of physics under the excitation of the control input. In this section, GTRS data is used to verify the control response of the tilt-rotor aircraft.
In the helicopter mode, the longitudinal stick in the hovering state is pushed forward by 5%, and the time response of the pitch angle and rate is shown in Figures 4 and 5. It can be seen from the figure that when the longitudinal stick is pushed forward and causes the aircraft head down, which conforms to the basic physical characteristics of the aircraft. In addition, the calculation model in this paper has the same trend and similar amplitude with the GTRS data.  In conclusion, the calculated values in this paper are in good agreement with those in the GTRS. In brief, the XV-15 tilt-rotor flight dynamics model is proved to be valid. So we have enough confidence in the following controller design.

L 1 Adaptive Controller Design
where xðtÞ = ½θ, q T is the pitch angle and pitch angle velocity, respectively; A m ∈ ℝ 2×2 is a known Hurwitz matrix whose characteristics meet the specifications of flight quality; b, c ∈ ℝ 2 is known constant vectors; uðtÞ ∈ ℝ is control input; ω is control efficiency; f ðt, xðtÞ, ξðtÞÞ is unknown nonlinear dynamic mapping; x ξ ðtÞ and ξðtÞ are the state and output of unmodeled dynamics, respectively, and gð⋅Þ and hð⋅Þ are nonlinear mappings of corresponding dimensions.           In this study, the uncertainties are mainly caused by aerodynamic parameters. They are always uniformly bounded and limited in how fast they can change. Thus, these assumptions are reasonable. The state prediction system, adaptive law, and control law of the L 1 adaptive control system of the tilt-rotor aircraft are designed as follows.

State Prediction System.
wherexðtÞ ∈ ℝ n is the state estimate of the system, and b ωðtÞ ∈ ℝ, b λðtÞ ∈ ℝ, b σðtÞ ∈ ℝ is the adaptive parameter estimates.

Control Law.
where rðsÞ is the Laplace transform of rðtÞ, k g ≜ −1/ðc ⊤ A −1 m bÞ, k is the design parameter of the low-pass filter, DðsÞ generally takes the form of integral, where b ηðsÞ is the Laplace transform of the following formula The adaptive control signal passes through the low-pass filter shown in the following equation before entering the actual system.
where CðsÞ is a filter that satisfies Cð0Þ = 1 and satisfies the following L 1 adaptive stability conditions. Theorem 4. For system (5), given a ρ 0 , there is ρ r > ρ in . When the system adopts the abovementioned state prediction system, adaptive law, and control law, the system is closed-loop stable when the following norm inequality is satisfied. where ρ r ðρ r Þ/ρ r d f x ð ρ r ðρ r ÞÞ, and ρ r ðρ r Þ ≜ max ½ρ r + γ 1 , L 1 ðρ r + γ 1 Þ + L 2 .

L 1 Adaptive Control System
Analysis. In order to analyze the closed-loop characteristics of the system, the following closed-loop reference system is defined where η ref ðtÞ ≜ f ðt, x ref ðtÞ, ξðtÞÞ.

Lemma 5.
When the closed-loop reference system satisfies the stability condition, for any τ > 0 x ref That is, the closed-loop reference system is BIBS stable.
Lemma 6. For system (5), when the system adopts the state prediction system, adaptive law, and control law and satisfies the stability condition (14), if the adaptive gain Γ is chosen to verify the lower bound Then, the following inequality holds Then, we have where γ 1 and γ 2 are dependent on f i ðt, xÞ bounds and L 1 controller design parameters. 7 International Journal of Aerospace Engineering Remark 1. As the adaptive gain Γ goes to infinity, the system (5) will follow the reference system (12) arbitrarily closely. It implies that the performance limitations are consistent with the hardware limitations.

Command System
Design. ADS-33E-PRF flight quality specification specifies a variety of response types. These response types are the guarantee for the helicopter to complete high-quality flight missions. The most basic is the angular rate response type and the attitude response type. In general, the aircraft basically has the angular rate response type through the preliminary stabilization system. In order to reduce the pilot workload, Attitude Command/Attitude Hold response type control system is designed for the longitudinal pitch channel.
According to ADS-33E-PRF flight quality specification, the flight quality constraint parameters of the attitude command type are the bandwidth and phase delay index for small-scale high-frequency attitude changes and the quickness index for medium attitude changes.      International Journal of Aerospace Engineering The following transfer function is used to design the pitch attitude command model.
According to the definition of bandwidth and phase delay in the ADS-33E-PRF flight quality specification, the relationship between the phase bandwidth ω BW of the attitude command model and the command model parameters ω, ζ, and τ SL are as follows.
Phase delay τ P where ω 180 is the frequency corresponding to equation (19) when the phase of the transfer function is 180 deg, which can be obtained by solving the following equation.
According to the corresponding relationship between response types and mission task recommended in the ADS-33E-PRF flight quality specification, the attitude response type is generally the response type used in complex situations such as degraded visual environment. At the same time, in the case of divided attention operation, the specification has made stricter requirements on the damping ratio of the medium period response. Therefore, the damping ratio of 1 is chosen. In this case, the quickness index of equation (19) is as follows Through the above analysis, the parameters in equation (19) could be determined by equations (20)-(23).

Simulations
The following parameters are selected through analysis. In order to examine the performance of the controller for tilt-rotor aircraft, three simulation scenarios are considered.    Figure 8. From the figure, it can be seen that the aircraft can track the reference signal well under the action of the control system. The control signal of the aircraft is shown in Figure 9. It can be seen from the figure the low-frequency components of the control signal flow into the real system under the action of the L 1 adaptive control system. The state prediction error is shown in Figure 10. It can be found that the prediction error quickly converges under the action of the L 1 adaptive control system.

Pitch Angle Capture Simulation in Airplane
Mode. In this case, the head-up 10 deg task in airplane mode is performed. Figure 11 shows the variations with time of the pitch rate and pitch angle for the experiment. From Figure 11(b), it can be seen that the pitch angle response can track the reference signal well. The control signal of the aircraft is shown in Figure 12. It can be seen from the figure that the frequency of the elevator signal is significantly reduced under the action of the L 1 adaptive control system. The state prediction error is shown in Figure 13, and it can be found that the prediction error quickly converges under the action of the L 1 adaptive control system.

Scenario 3: Automatic Conversion
Simulation. Conversion mode is the most important flight mode of tilt-rotor aircraft, which has a great impact on flight safety. In order to further verify the performance of the flight control system, an automatic conversion simulation was implemented. The nacelles were assumed to rotate with a constant angular speed, as shown in Figure 14. The pitch angle command is shown in Figure 15(a) which is determined by the trim  11 International Journal of Aerospace Engineering characteristics. As we can know from Figure 15, the pitch angle response can track the reference signal well and the pitch rate changes smoothly. The change curves of its velocity and height are shown in Figure 16. It can be seen that it is an accelerated process with a certain altitude climb from helicopter mode to airplane mode. From Figure 17, the conversion path is completely in the conversion corridor, which meets the flight condition constraints of the conversion mode. The control signal of the aircraft is shown in Figure 18. It can be seen from the figure that the signals of the longitudinal cycle pitch and elevator could be physically implemented. In general, under the action of the L 1 adaptive control system, the states and control signals in the whole conversion process are reasonable.
In this section, three sets of simulation tests have been completed. It can be obtained from the first set simulation that the control method in this paper can track pitch commend in different flight modes, while MRAC cannot complete this simulation task. The second test shows that the control method in this paper can achieve good flight quality, and the control signal can be physically realized. The following conclusion can be drawn from the third set of tests is that the proposed method can realize automatic transition flight.
In addition, in the three sets of simulation tests, exactly the same controller architecture and parameter setting are selected, which avoids the repeated design of the control law and reduces the workload of the control system design.

Conclusions
In this paper, a L 1 adaptive controller for tilt-rotor aircraft has been proposed. The effectiveness of the controller is verified through simulation. The main conclusions are summarized as follows.
(1) Tilt-rotor aircraft has three flight modes in the flight envelope, which results in a large uncertainty in the flight dynamics model. Considering the modeling uncertainty, tilt-rotor aircraft is described as a form with unmodeled dynamics, and the L 1 adaptive control system scheme is formed (2) The MRAC method was compared to verify the effectiveness of the proposed method. The controller in this paper can track pitch commend in different flight modes, while MRAC task failed (3) The simulation test of the conversion flight is completed with the same controller architecture and parameters setting. The conversion from helicopter mode to airplane mode is successfully realized, which significantly reduced the workload of control law design (4) In the future work, we will plan to implement the designed algorithm in practical application by conducting scientific experiments

Data Availability
For data availability, if the researcher needs data of this manuscript, the corresponding author can provide the data upon request.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.

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International Journal of Aerospace Engineering